Complex Geometry

Course Overview

This graduate-level course offers an introduction to the fundamental concepts and techniques of complex differential geometry.

The central aim of the course is to understand the criteria that determine when a compact complex manifold can be realized as a smooth projective algebraic variety. This is the celebrated Kodaira Embedding Theorem, a cornerstone result that provides a precise differential-geometric condition (the existence of a positive line bundle, or a Hodge metric) for a complex manifold to be projective (and thus algebraic by Chow's theorem). We will work through the necessary machinery to fully prove this theorem.

Time permitting, we will then discuss Kodaira-Spencer deformation theory and discuss the case of Calabi-Yau manifolds, studied by Tian-Todorov.

Prerequisites: Complex analysis and differential geometry. Familiarity with Riemannian geometry and vector bundles is desirable.

Notes and references

I will draw from

• D. Huybrechts, Complex Geometry: An Introduction


• J.-P. Demailly, Complex Analytic and Differential Geometry

I will (try to) keep up-to-date lecture notes as the term progresses, available here.

Syllabus

0. Overview


1. Holomorphic functions: Extendability & rigidity


2. Complex and almost complex manifolds


3. Vector bundles and sheaves


4. Kodaira dimension and Siegel's theorem


5. Divisors and blow-ups


6. Metrics and connections


7. The Kähler condition


8. Positivity and vanishing


9. The Kodaira embedding theorem


10. Kodaira-Spencer deformation theory


11. (Formal) Tian-Todorov theorem

Exercises

There will be a list of exercises published weekly.